Manfred Hammer, Samer Alhaddad, and Jens Förstner, "Hybrid coupled-mode modeling in 3D: perturbed and coupled channels, and waveguide crossings," J. Opt. Soc. Am. B 34, 613-624 (2017)

The 3D implementation of a hybrid analytical/numerical variant of the
coupled-mode theory is discussed. Eigenmodes of the constituting
dielectric channels are computed numerically. The frequency-domain
coupled-mode models then combine these into fully vectorial
approximations for the optical electromagnetic fields of the composite
structure. Following a discretization of amplitude functions by 1D
finite elements, procedures from the realm of finite-element numerics
are applied to establish systems of linear equations for the
then-discrete modal amplitudes. Examples substantiate the functioning
of the technique and allow for some numerical assessment. The full 3D
simulations are highly efficient in memory consumption, moderately
demanding in computational time, and, in regimes of low radiative
losses, sufficiently accurate for practical design. Our results
include the perturbation of guided modes by changes of the refractive
indices, the interaction of waves in parallel, horizontally or
vertically coupled straight waveguides, and a series of crossings of
potentially overlapping channels with fairly arbitrary relative
positions and orientations.

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Computational effort spent on simulations of the three crossing
configurations of Fig. 12 by the present 3D HCMT scheme and by the
frequency-domain solver of the CST Microwave Studio [37]. Observed data for
peak memory use and total program runtime are compared; the
extension of the computational intervals [HCMT, cf. Fig. 2(e)] and the volume of
the computational domain (CST) are listed. In both cases,
machines with Intel 16-core Xeon CPUs (2.9 GHz) with 128 GB of
memory were used. The commercial CST software (MS Windows
Server) ran in parallel on up to 8 cores, depending on the
phase of the computations, while our HCMT code (Linux, g++)
occupied a single core only.

Table 2.

Phase Shifts Due to Waveguide Core Perturbations^{a}

$\mathrm{\Delta}{n}_{\mathrm{eff}}$

$\mathrm{\Delta}n=0.1$

$\mathrm{\Delta}n=0.2$

TE

TM

TE

TM

HCMT

0.075

0.049

0.154

0.100

JCMwave

0.078

0.051

0.162

0.110

Shifts $\mathrm{\Delta}{n}_{\mathrm{eff}}$, as predicted by HCMT
simulations of a waveguide segment with its core
refractive index increased by $\mathrm{\Delta}n$. Entries
“JCMwave” (reference) are computed by direct
mode analysis [22]
of the waveguide with the perturbed core.

Coupling lengths ${L}_{\mathrm{c}}$ for horizontally (a) and
vertically (b) coupled waveguides with the same parameters
as in Fig. 3, as
determined by the HCMT formalism, and via a direct
supermode analysis with the JCMwave solver [22] (reference).

Table 4.

Perpendicular Waveguide Crossing, Transmittance and
Reflectance^{a}

For the perpendicular waveguide crossing of Section 3.D: relative
guided power transferred to the polarized modal outlet
straight ahead ($\to $), reflected
($\leftarrow $), and directed toward the
lateral outlets ($\uparrow ,\downarrow $), for TE- (a) and
TM-polarized excitation (b). Reference calculations using
the CST solver [37].

Tables (4)

Table 1.

Computational Effort for HCMT Simulations and Full Wave Reference
Calculations^{a}

Computational effort spent on simulations of the three crossing
configurations of Fig. 12 by the present 3D HCMT scheme and by the
frequency-domain solver of the CST Microwave Studio [37]. Observed data for
peak memory use and total program runtime are compared; the
extension of the computational intervals [HCMT, cf. Fig. 2(e)] and the volume of
the computational domain (CST) are listed. In both cases,
machines with Intel 16-core Xeon CPUs (2.9 GHz) with 128 GB of
memory were used. The commercial CST software (MS Windows
Server) ran in parallel on up to 8 cores, depending on the
phase of the computations, while our HCMT code (Linux, g++)
occupied a single core only.

Table 2.

Phase Shifts Due to Waveguide Core Perturbations^{a}

$\mathrm{\Delta}{n}_{\mathrm{eff}}$

$\mathrm{\Delta}n=0.1$

$\mathrm{\Delta}n=0.2$

TE

TM

TE

TM

HCMT

0.075

0.049

0.154

0.100

JCMwave

0.078

0.051

0.162

0.110

Shifts $\mathrm{\Delta}{n}_{\mathrm{eff}}$, as predicted by HCMT
simulations of a waveguide segment with its core
refractive index increased by $\mathrm{\Delta}n$. Entries
“JCMwave” (reference) are computed by direct
mode analysis [22]
of the waveguide with the perturbed core.

Coupling lengths ${L}_{\mathrm{c}}$ for horizontally (a) and
vertically (b) coupled waveguides with the same parameters
as in Fig. 3, as
determined by the HCMT formalism, and via a direct
supermode analysis with the JCMwave solver [22] (reference).

Table 4.

Perpendicular Waveguide Crossing, Transmittance and
Reflectance^{a}

For the perpendicular waveguide crossing of Section 3.D: relative
guided power transferred to the polarized modal outlet
straight ahead ($\to $), reflected
($\leftarrow $), and directed toward the
lateral outlets ($\uparrow ,\downarrow $), for TE- (a) and
TM-polarized excitation (b). Reference calculations using
the CST solver [37].